I am not sure I understand your terminology: "These count arithmetic sequences of all possible lengths and starting points" (my emph) is this just a typo for difference? Also do I understand correctly you insist on a fixed difference. And what do you mean by an explicit proof?
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quidNov 23 '12 at 21:36

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If you're asking what I think you're asking: Compute $f(r,\delta)$, the maximum possible density of 3-term progressions with difference $r$ in a set of density $\delta$, I think the answer is fairly easy. It's just $\delta$. For example, insert blocks of length $N\gg r$ spaced regularly $N/\delta$ apart. This sequence has density $\delta$ and most elements of the set are initial points of 3-term progressions spaced $r$ apart. You clearly can't do better than this.
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Anthony QuasNov 23 '12 at 22:25

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@Anthony Quas: from the context a lower bound seems more relevant. But I agree that it is not quite clear what is asked for. And as implict in my comment I somehow doubt that with fixed difference there is much to say (with arbitrary difference this would be quite different); as your example shows, as other sets with same density would have no AP of that difference.
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quidNov 23 '12 at 23:41

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You can achieve density $2/3$ with no APs of common difference $r$ by knocking out every third element of each of the $r$ infinite APs of common difference $r$. And as Anthony Quas points out in the comments, for general $\delta$ we can achieve a density $\delta$ of all APs of common difference $r$ by taking our set to be a union of long intervals. So we can't ask for too much about the common differences of the APs we obtain from Roth's theorem.